When a power transformer on an electrical transmission or distribution network is re-energized, it is known that transient over-currents can occur owing to a difference between the residual flux values in each part of the magnetic circuit and the fluxes generated by the voltages imposed at the terminals of each winding.
These over-currents rich in harmonics may, in some configurations of the network, exhibit values much higher than the transformer's permissible levels.
In addition, these over-currents may create major electro-dynamic forces at the windings, leading to accelerate the degradation of the transformer (deformation, winding displacement).
These problems of over-currents and over-voltages can also be encountered in other electrical devices including a magnetic circuit and electric windings (start-up of electrical machines).
As an illustration, attention is given below to the case of a single-phase transformer.
Before energizing, the flux φ in the ferromagnetic material forming the magnetic circuit has a value φr called residual flux.
This residual flux is dependent on the de-energizing conditions of the transformer, which are generally not controlled, on the type of magnetic circuit (e.g. its geometry) and on the intrinsic parameters of its constituent material.
This residual flux is likely to develop over time, in particular on account of outside stresses which may be exerted upon the de-energized transformer (e.g. under the influence of electrical devices in the vicinity of the transformer).
At the time of energizing at t=0, since the applied voltage is an alternating voltage, the voltage at the terminals of the inductor winding can be written as: V0=V√{square root over (2)} cos(α)
where:                V is the root mean square of the imposed voltage;        α is the angle representing the phase at the time of energizing.        
V0 therefore has a value that is solely dependent on α.
To this value there is a corresponding flux φ0 imposed within the magnetic circuit. The operating equation is therefore the following:
      V    ⁢          2        ⁢          cos      ⁡              (                              ω            ⁢                                                  ⁢            t                    +          α                )              =            Ri      ⁡              (        t        )              +          n      ⁢                        ⅆ                      ϕ            ⁡                          (              t              )                                                ⅆ          t                    
where:                ω is the voltage pulse        R is the total resistance of the electric circuit including that of the inductor winding        n is the number of turns of the inductor winding        φ is the mean flux within the magnetic circuit.        
It is known that the expression of flux, with some approximations, is the following:
      ϕ    ⁡          (      t      )        =                              V          ⁢                      2                                    n          ⁢                                          ⁢          ω                    ⁢              sin        ⁡                  (                                    ω              ⁢                                                          ⁢              t                        +            α                    )                      +                  (                              ϕ            r                    -                                                    V                ⁢                                  2                                                            n                ⁢                                                                  ⁢                ω                                      ⁢            sin            ⁢                                                  ⁢            α                          )            ⁢              ⅇ                              -            t                    ⁢                      /                    ⁢          τ                    
with:                τ=L/R and L is the inductance of the inductive winding.        
It is then possible to determine the current i(t) as a function of the curve B(H) of the magnetic material of the circuit.
Optimal energizing of the transformer takes place at a given angle α such that the transient flux (and hence transient current) i.e. the maximum current reached after energizing, is as low as possible in order to protect the transformer.
For example, if φr=0 and α=0 (i.e. energizing at maximum voltage and no residual flux), then:
      ϕ    ⁡          (      t      )        =                    V        ⁢                  2                            n        ⁢                                  ⁢        ω              ⁢          sin      ⁡              (                  ω          ⁢                                          ⁢          t                )            
which means that there is no inrush current. Energizing is therefore optimal.
On the other hand, if φr=φr max and α=3π/2 (i.e. energizing at 0 voltage and maximum residual flux), then:
      ϕ    ⁡          (      t      )        =                    -                              V            ⁢                          2                                            n            ⁢                                                  ⁢            ω                              ⁢      cos      ⁢                          ⁢              (                  ω          ⁢                                          ⁢          t                )              +                  (                              ϕ                          r              ⁢                                                          ⁢              max                                +                                    V              ⁢                              2                                                    n              ⁢                                                          ⁢              ω                                      )            ⁢              ⅇ                              -            t                    ⁢                      /                    ⁢          τ                    
In this case the flux reaches very high values leading to high inrush current or causes major temporary harmonic over-voltages on the network.
These two examples show the advantage of having knowledge of the value of residual flux.
One known solution for evaluating residual flux is based on the fact that voltage is homogeneous to flux derivation and therefore consists of evaluating residual flux by integrating the voltage at the terminals of the transformer before it is de-energized.
Said method is described for example in document US 2010/0013470.
Documents DE 196 41 116 and DE 36 14 057 also disclose methods using data on the state of the device before it is de-energized to estimate an optimal energizing time.
However, said indirect method for determining residual flux may, in some configurations of the electric network supplying the transformer, prove to be scarcely precise and scarcely robust since phenomena may have occurred which change the magnetic state of the magnetic circuit, and imprecision in measurement of voltage—which is the input data for calculating flux—makes this calculation little accurate (offset, drift, low voltage level, noisy signal).
Additionally, a long time may elapse between the de-energizing and energizing of a transformer, which requires the saving of data over a long period and regular measurement of flux to verify changes thereof.
It is one objective of the invention therefore to allow more precise, simple and reliable controlling of the switching time of a transformer or of any device comprising a magnetic circuit and one or more conductive windings through which a current passes when in operation, such as a rotating machine for example.
A further objective of the invention is to provide a simple and reliable method for energizing a transformer under optimal conditions.
A further objective of the invention is to design a system for determining residual flux in a magnetic circuit which provides better performance and is more precise than current systems and is easy to implement.